White light achromatic grating imaging polarimeter

ABSTRACT

White-light snapshot channeled linear imaging (CLI) polarimeters include polarization gratings (PGs) configured to produce a compensated shear between portions of an input light flux in first and second polarization states. The disclosed CLI polarimeters can measure a 2-dimensional distribution of linear Stokes polarization parameters by incorporating two identical PGs placed in series along an optical axis. In some examples, CLI polarimeters are configured to produce linear (S 0 , S 1 , and S 2 ) and complete (S 0 , S 1 , S 2  and S 3 ) channeled Stokes images.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 13/225,315, filed Sep. 2, 2011, which claims the benefit ofU.S. Provisional Application 61/402,767, filed Sep. 3, 2010. Thisapplication also claims the benefit of U.S. Provisional Application61/463,488, filed Feb. 17, 2011. All of the above applications areincorporated herein by reference.

FIELD

The disclosure pertains to imaging polarimeters using polarizinggratings.

BACKGROUND

Polarization images can yield higher contrast than intensity images,providing the opportunity for dramatically improved objectidentification. Furthermore, incorporation of a polarimeter into adetection system allows for the potential to ascertain the Stokesparameter elements of a scene, thereby giving a complete identificationof the polarization state of light reflected or emitted from objects inthe scene. From such an analysis, the spatially varying two-dimensionalstate of polarization (SOP) can be determined.

SOP analysis is a useful technique for object characterization anddistinction, particularly for differentiating man made versus naturalobjects. This is particularly valuable in the thermal infrared; ifobjects in a scene are emitting close to the background temperature ofthe environment (i.e., they are close to thermal equilibrium with theirenvironment), then thermal detection typically yields ambiguous results.Addition of polarimetry data can often significantly enhance images ofsuch objects as polarimetry can supply information that is unavailableby intensity imaging. For example, typical long-wavelength infrared(LWIR) intensity images provide little indication of the presence of avehicle in the shadows of tree, while a polarization image makes thepresence of an automobile obvious due to polarization associated withthe smooth surfaces of the automobile.

Current techniques for imaging polarimetry include rotating retarderpolarimeters. Through a series of sequential measurements, the completespatial distribution of Stokes parameters in a scene can be determined.This method has several significant limitations. Rotating parts can leadto vibrational and mechanical problems. Images of dynamic scenes canalso contain polarization artifacts as a result of combining a series ofmeasurements. Other problems are related to oversampling and spatialsynchronization.

Some of the problems with rotating retarder imaging polarimetry can beaddressed with “snapshot” systems that do not require dynamiccomponents, but instead take advantage of spatial carrier fringes andFourier reconstruction techniques in order to provide a completepolarization analysis of a scene. Examples of such approaches aredescribed in Oka and Saito, “Snapshot complete imaging polarimeter usingSavart plates,” Proc. SPIE 6295:629508 (2008) and Oka and Kaneko,“Compact complete imaging polarimeter using birefringent wedge prisms,”Opt. Exp. 11:1510-1519 (2003), both of which are incorporated herein byreference. These approaches use birefringent materials to producepolarization dependent phase differences to produce snapshot images.

One example of such a snapshot system is based on a pair of Savartplates (SPs) introduced in a collimated space in an imaging system. AnSP shears incident radiation using crystal birefringence to produce twolaterally displaced, orthogonally polarized beams. By combining twoorthogonal SPs, an incident optical flux is sheared to create fourseparate beams. After transmission by an analyzer, these beams arerecombined with a lens, resulting in amplitude modulated interferencefringes containing state of polarization (SOP) information on the imageplane.

While such SP systems are impressive in their snapshot capabilities,they suffer from significant limitations. Due to the reliance oninterference effects, the temporal coherence of imaging radiationpresents a constraint in that the visibility of the interference fringesis inversely proportional to the spectral bandwidth. For instance, inthe LWIR (8-12 μm wavelengths), a fringe visibility of 50% at a meanwavelength of 10 μm requires limiting optical bandwidth Δλ_(50%)≈373 nm,which is a significant constraint with respect to the signal to noiseratio (SNR) of the acquired data. In addition, SP polarimeters requireSPs which can be expensive due to the birefringent crystals required. Inmany wavelength regimes, especially the infrared, the required largecrystals (clear apertures>25 mm with thicknesses>10 mm) are eitherunavailable or prohibitively expensive. Moreover, materials suitable forLWIR use such as CdSe or CdS have birefringences B=|n_(e)−n_(o)| thatare approximately 10 times less than those of materials suitable for useat visible wavelengths. As a result, thick crystals are needed.

These birefringent material limitations can be avoided through theimplementation of a reflective interferometric scheme. Mujat et. al.,“Interferometric imaging polarimeter,” JOSA A:21:2244-2249 (2004), whichis incorporated herein by reference, discloses an interferometricimaging polarimeter based on a modified Sagnac interferometer. In thissystem, a polarizing beam splitter is used to transmit an input beaminto an interferometer, and a phase difference between orthogonalpolarizations produced by displacing one of the mirrors in theinterferometer is used to create an interference pattern. Irradiancemeasurements and coherence matrix techniques are then employed todetermine the state of polarization from a set of two temporally spacedimages. These methods are subject to similar registration problems thatplague rotating retarder polarimeters for dynamic scenes.

SUMMARY

White light polarization grating based imaging polarimeters andassociated methods are disclosed herein. In some examples, so-called“snapshot imaging polarimeters” are described that operate over broadwavelength ranges, including thermal infrared wavelengths and visibleoptical wavelengths. Polarizing diffraction gratings are situated toproduce a shear between first and second polarization components of areceived radiation flux that is proportional to wavelength so thatwhite-light broadband interference fringes are produced. In someexamples, complete polarization data for a scene of interest is producedas all four Stokes parameters, while in other examples, only one orseveral polarization characteristics are determined such as one or moreof the Stokes parameters. Stokes parameters are encoded onto a sequenceof one or two dimensional spatial carrier frequencies so that a Fouriertransformation of a generated fringe pattern enables reconstruction ofthe Stokes parameter distribution.

In one example, an apparatus comprises at least one polarizing gratingconfigured to produce a dispersion compensated shear between portions ofan input light flux associated with first and second polarizations. Adetector is situated to receive an output light flux corresponding to acombination of the sheared first and second portions of the input lightflux and produce an image signal. An image processor is configured toproduce a polarization image based on the image signal. Typically, apolarization analyzer is situated between the at least one grating andthe detector. In other examples, the at least one polarizing gratingincludes two polarizing gratings, wherein the first grating is situatedto produce a first shear portion by directing the first polarizationalong a first direction and the second polarization along a seconddirection, and the second grating is situated to produce a second shearportion by directing the first polarization directed by the firstgrating along the second direction and the second polarization directedby the first grating along the first direction. In other examples, theat least one polarizing grating includes a first pair and a second pairof polarizing gratings configured to produce dispersion compensatedshear along a first axis and a second axis. Typically, the detector isan array detector. In some examples, the image processor is configuredto produce the polarization image based on an amplitude modulation ofinterference fringes. In further examples, the image processor isconfigured to select at least one spatial frequency component of therecorded image signal and determine an image polarization characteristicbased on an intensity modulation associated with an image signalvariation at the selected spatial frequency. Representative imagepolarization characteristics include one or more or a combination ofStokes parameters S₀, S₁, S₂, and S₃. In some alternatives, thepolarizing gratings are blazed birefringent gratings and/or liquidcrystal gratings. In typical examples, the dispersion compensated shearis proportional to a separation between the first grating and the secondgrating. In some embodiments, the first grating is situated to directthe first polarization in a + diffraction order and the secondpolarization in a − diffraction order, and the second grating isconfigured to effectively direct the first polarization in a −diffraction order and the second polarization in a + diffraction orderso as to produce the dispersion compensated shear. In other embodiments,the first grating is situated to direct the first polarization above,and away from, the optical axis and the second polarization below, andaway from, the optical axis, and the second grating is configured, in anappropriate manner, to redirect the first and second polarizations topropagate parallel with the optical axis, so as to produce thedispersion compensated shear.

Representative methods comprise receiving an input optical flux andproducing a shear between first and second portions of the input opticalflux associated with first and second states of polarization that isproportional to a wavelength of the input optical flux by directing thefirst and second portions to a pair of polarizing gratings. Apolarization characteristic of the input optical flux is estimated basedon a spatial frequency associated with the shear in an intensity patternobtained by combining the sheared first and second portions of the inputoptical flux. In some examples, each of the first and second portions ofthe incident optical flux are directed to the at least one diffractiongrating so as to produce a shear having a magnitude associated with agrating period. In some examples, the shear is inversely proportional toa grating period and directly proportional to a grating order.Typically, the first and second portions are combined with at least onefocusing optical element of focal length f, wherein the spatialfrequency is inversely proportional to f.

Representative imaging polarimeters include a first polarizing gratingconfigured to diffract portions of an input light flux having a firststate of polarization and a second state of polarization in a firstdirection and a second direction, respectively. A second polarizinggrating is configured to receive the diffracted portion from the firstpolarizing grating and diffract the portions associated with the firststate of polarization and the second state of polarization along thesecond direction and the first direction, respectively, so that thefirst and second portions propagate displaced from and parallel to eachother. A polarization analyzer is configured to produce a common stateof polarization of the first and second portions, and a focusing elementis configured to combine the first and second portions. A detector isconfigured to receive the intensity pattern and produce a detectedintensity pattern. In some examples, an image processor is configured toproduce a polarization image based on the detected intensity pattern. Inother examples, the detected intensity pattern is associated with ashear produced by the displacement of the first and second portions.

The foregoing and other features, and advantages of the disclosedtechnology will become more apparent from the following detaileddescription, which proceeds with reference to the accompanying figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a modified Sagnac interferometer configured toproduce shear between counter-propagating optical fluxes using twodiffraction gratings.

FIG. 1B is an unfolded view of a portion of the interferometer of FIG.1A.

FIG. 2 illustrates an interferometer-based polarimeter configured thatincludes an input quarter wave retarder and configured for estimation ofspatial distributions of Stokes parameters S₀, S₁, and S₂.

FIG. 3 illustrates an interferometer-based polarimeter that includes aninput quarter wave retarder and output linear analyzer configured forestimation of spatial distributions of linear polarization.

FIGS. 4A-4B illustrate propagation of multiple spectral components in adispersion compensated interferometer that includes two blazed gratings.

FIG. 5 is a graph of theoretical diffraction efficiency for a blazedgrating designed for a wavelength of roughly 8 μm on a ZnSe substrate.Diffraction efficiencies for the 0, +/−1, +/−2, and +/−3 orders areshown.

FIG. 6 is a graph of diffraction efficiency for a multiple order “deep”blazed diffraction grating having a 2.12 μm grating depth.

FIGS. 7A-7B illustrate single order and multiple order blazed gratings.

FIG. 8A illustrates a Sagnac interferometer based imaging polarimeterthat includes multiple-order blazed gratings (MBGs) situated to providemultiple diffraction orders in two directions.

FIG. 8B illustrates a grating assembly that provides multiplediffraction orders in a plurality of directions.

FIG. 9 illustrates a Sagnac interferometer system that includesdiffraction gratings formed on mirror surfaces.

FIG. 10 illustrates determination of an optical path difference (OPD)associated with shear.

FIGS. 11A-11B illustrate polarimeters based on parallel or serialarrangements of Sagnac interferometers.

FIG. 12 is a schematic diagram of blazed birefringent grating pairssituated to produce a compensated shear.

FIG. 13 is a is a schematic diagram of two pairs of blazed birefringentgratings configured to produce four beams based on compensated shears inan X-direction and a Y-direction.

FIG. 14 illustrates production of a shear with a polarizing gratingpair.

FIG. 15 illustrates production of a shear with polarizing liquid crystalgratings.

FIG. 16 is a schematic diagram of a CLI polarimeter using right circular(RC) and left circular (LC) polarizations diffracting into the −1 and +1diffraction orders, respectively.

FIG. 17 is an arrangement for establishing the measurement accuracy of aCLI polarimeter in white-light.

FIGS. 18A-18C are white-light interference fringe patterns generated ina central 100×100 pixels on a focal plane array at polarizerorientations of θ=0°, θ=50°, and θ=90°, respectively.

FIG. 19 is a graph comparing measured and theoretical polarimetricreconstructions.

FIG. 20 includes measured zero-order and total first-order(T_(±1)=T₊₁+T⁻¹) transmission spectra of a representative polarizationgrating.

FIG. 21 is a schematic diagram of an experimental setup for viewingoutdoor targets with a CLI polarimeter. An afocal telescope is includedto allow the scene to be defocused while maintaining focus on theinterference fringes.

FIG. 22 is a raw image of a moving vehicle prior to extraction of Stokesparameters. Interference fringes are located in areas of the scene thatare linearly polarized and are particularly evident in the vehicle hood.

FIGS. 23A-23D are images obtained from the polarization datacorresponding to the image of FIG. 22, wherein the images are based onS₀, degree of linear polarization (DOLP), S₁/S₀ and S₂/S₀, respectively.

FIG. 24A is a schematic of a full imaging Stokes polarimeter that canprovide images based on all four Stokes parameters. Polarizationgratings PG₁ and PG₂ diffract in the yz plane, while polarizationgratings PG₃ and PG₄ diffract in the xz plane.

FIG. 24B is a perspective view of the polarimeter of FIG. 24A.

DETAILED DESCRIPTION

As used in this application and in the claims, the singular forms “a,”“an,” and “the” include the plural forms unless the context clearlydictates otherwise. Additionally, the term “includes” means “comprises.”Further, the term “coupled” does not exclude the presence ofintermediate elements between the coupled items.

The systems, apparatus, and methods described herein should not beconstrued as limiting in any way. Instead, the present disclosure isdirected toward all novel and non-obvious features and aspects of thevarious disclosed embodiments, alone and in various combinations andsub-combinations with one another. The disclosed systems, methods, andapparatus are not limited to any specific aspect or feature orcombinations thereof, nor do the disclosed systems, methods, andapparatus require that any one or more specific advantages be present orproblems be solved. Any theories of operation are to facilitateexplanation, but the disclosed systems, methods, and apparatus are notlimited to such theories of operation.

Although the operations of some of the disclosed methods are describedin a particular, sequential order for convenient presentation, it shouldbe understood that this manner of description encompasses rearrangement,unless a particular ordering is required by specific language set forthbelow. For example, operations described sequentially may in some casesbe rearranged or performed concurrently. Moreover, for the sake ofsimplicity, the attached figures may not show the various ways in whichthe disclosed systems, methods, and apparatus can be used in conjunctionwith other systems, methods, and apparatus. Additionally, thedescription sometimes uses terms like “produce” and “provide” todescribe the disclosed methods. These terms are high-level abstractionsof the actual operations that are performed. The actual operations thatcorrespond to these terms will vary depending on the particularimplementation and are readily discernible by one of ordinary skill inthe art.

As used herein, an optical flux refers to electromagnetic radiation in awavelength range of from about 100 nm to about 100 μm. In some examples,an optical flux has a spectral width that can be as large as 0.5, 1, 2,5, or 10 times a center wavelength, or can comprises a plurality ofspectral components extending over similar spectral bandwidths. Suchoptical fluxes can be referred to as large bandwidth optical fluxes.Typically, an optical flux is received from a scene of interest andamplitude, phase, spectral, or polarization modulation (or one or morecombinations thereof) in the received optical flux is processed based ona detected image associated with a spatial variation of the optical fluxwhich can be stored in one or more computer-readable media as an imagefile in a JPEG or other format. In the disclosed examples, so-called“snapshot” imaging systems are described in which image data associatedwith a plurality of regions or locations in a scene of interest(typically an entire two dimensional image) can be obtained in a singleacquisition of a received optical flux using a two dimensional detectorarray. However, images can also be obtained using one dimensional arraysor one or more individual detectors and suitable scanning systems. Insome examples, an image associated with the detected optical flux isstored for processing based on computer executable instruction stored ina computer readable medium and configured for execution on generalpurpose or special purpose processor, or dedicated processing hardware.In addition to snapshot imaging, sequential measurements can also beused. For convenience, examples that provide two dimensional images aredescribed, but in other examples, one dimensional (line) images orsingle point images can be obtained.

For convenience, optical systems are described with respect to an axisalong which optical fluxes propagate and along which optical componentsare situated. Such an axis is shown as bent or folded by reflectiveoptical elements. In the disclosed embodiments, an xyz-coordinate systemis used in which a direction of propagation is along a z-axis (which mayvary due to folding of the axis) and x- and y-axes define transverseplanes. Typically the y-axis is perpendicular to the plane of thedrawings and the x-axis is perpendicular to the y-axis and the z-axisand is in the plane of the drawings.

In representative examples, the imaging polarimetry methods andapparatus disclosed herein can be used to estimate a 2-dimensionalspatial Stokes parameter distribution of a scene in order tocharacterize aerosol size distributions, distinguish manmade targetsfrom background clutter, evaluate distributions of stress birefringencein quality control, evaluate biological tissues in medical imaging, orfor other purposes. While in typical examples, image data is evaluatedso as to correspond to one or more components of a Stokes vector, datacan be processed to obtain other polarization characteristics such asellipticity or can be based on other representations such as thoseassociated with Jones matrices.

In the disclosed embodiments, interferometers are configured to includediffraction gratings so as to produce a shear between orthogonallypolarized components of an input optical flux that is proportional to awavelength of the input optical flux. For large bandwidth opticalfluxes, shear for each spectral component is proportional to awavelength of the spectral component. A shear between optical fluxesthat varies linearly with flux wavelength is referred to herein as adispersion-compensated shear. In some examples, polarimeters includeoptical systems that can provide a total shear that includes adispersion compensated shear and a dispersive shear. As discussed below,a dispersion compensated shear is associated with interference patternshaving amplitude modulations at a spatial frequency that is independentof optical wavelength.

Polarization properties of a scene can be conveniently described using aStokes vector. A scene Stokes vector S(x,y), is defined as:

$\begin{matrix}{{{S\left( {x,y} \right)} = {\begin{bmatrix}{S_{0}\left( {x,y} \right)} \\{S_{1}\left( {x,y} \right)} \\{S_{2}\left( {x,y} \right)} \\{S_{3}\left( {x,y} \right)}\end{bmatrix} = \begin{bmatrix}{{I_{0}\left( {x,y} \right)} + {I_{90}\left( {x,y} \right)}} \\{{I_{0}\left( {x,y} \right)} - {I_{90}\left( {x,y} \right)}} \\{{I_{45}\left( {x,y} \right)} - {I_{135}\left( {x,y} \right)}} \\{{I_{R}\left( {x,y} \right)} - {I_{L}\left( {x,y} \right)}}\end{bmatrix}}},} & (1)\end{matrix}$

wherein x, y are spatial coordinates in the scene, S₀ is the total powerof the beam, S₁ denotes a preference for linear polarization at 0° overlinear polarization at 90°, S₂ denotes a preference for linearpolarization at 45° over linear polarization at 135°, S₃ denotes apreference for right circular over left circular polarization states,and I(x,y) refers to optical flux intensity. By measuring all fourelements of S(x,y), a complete spatial distribution of the polarizationstate associated with an scene can be determined. The Stokes vectorpermits assessment of partially polarized optical fluxes anddetermination of an extent of polarization as, for example,

$\frac{\left( {S_{1}^{2} + S_{2}^{2} + S_{3}^{2}} \right)^{1/2}}{S_{0}}.$

As discussed above, some conventional approaches to measuring sceneStokes parameters are based on recording multiple intensity measurementssequentially using different configurations of polarization analyzers.The Stokes parameters can then be calculated using Mueller matrices.However, time-sequential measurements of a rapidly changing scene aresusceptible to temporal misregistration. The disclosed methods andapparatus can reduce or eliminate such misregistration errors byacquiring scene image data in a single snapshot. Sequential measurementscan be made as well, if desired.

According to representative examples, interferometrically generatedcarrier frequencies are amplitude modulated with spatially-dependent2-dimensional Stokes parameters associated with a scene to be imaged.Such methods can be referred to as channeled image polarimetry (CIP)methods. In typical examples, all the Stokes parameters are directlymodulated onto coincident interference fringes so that misregistrationproblems are eliminated, and images can be acquired with readilyavailable lenses and cameras.

Example 1 Symmetric Grating Based Embodiments

For convenient illustration, representative embodiments are described inwhich diffraction gratings are symmetrically situated in a Sagnacinterferometer with respect to reflectors that definecounter-propagating optical paths. Following this description, otherexamples with arbitrary grating placements are described.

With reference to FIG. 1A, a representative Sagnac interferometer 100includes a polarizing beam splitter (PBS) 102, and reflective surfaces104, 106 that define an interferometer optical path 108. Forconvenience, the path 108 is also referred to as an interferometer axisherein. As shown in FIG. 1A, the interferometer axis 108 is folded bythe reflective surfaces 104, 106. Blazed transmission gratings (BGs)110, 112, are situated along the axis 108 at an axial distances b₁, b₂from the reflective surfaces 106, 104, respectively. The PBS 102 isconfigured to receive an input optical flux 116 that is directed alongthe axis 108 so that portions of the input optical flux 116 arereflected or transmitted to respective reflective surfaces 104, 106 andthe associated BGs 110, 112. As shown in FIG. 1A, the reflected andtransmitted portions of the input optical flux counter-propagate in theinterferometer 100. Typically, the input flux 116 is a collimatedoptical flux associated with an image scene, and a lens 118 is situatedto receive and combine the counter-propagating portions of the inputoptical flux received from the PBS 102 after transmission by apolarization analyzer 131.

The PBS 102 can be a thin-film based beam splitter such as a polarizingbeam splitter cube, a wire grid beam splitter (WGBS), or otherpolarization dependent beam splitter. The blazed diffraction gratingscan be ruled gratings, holographic gratings, or other types of gratings.Reflective surfaces such as the surfaces 104, 106 can be provided asmetallic coatings, polished metal surfaces, dielectric coatings, orbased on total internal reflection. As shown in FIG. 1A, the reflectivesurfaces 104,106 are provided by respective mirrors 105, 107.

The input optical flux 116 is divided into orthogonal polarizationcomponents by the polarizing beam splitter 102 and the components aredirected along respective arms of the interferometer 100. For example,the portion of the light flux 116 transmitted by the PBS 102 is directedalong the axis 108 to the diffraction grating 112 to the reflectivesurface 106. As shown in FIG. 1A, the reflective surface 106 is situateda distance b₁ from the BG 112 measured along the axis 108. Thediffraction grating 112 diffracts at least a portion of the incidentflux into a single diffraction order at an angle θ, given by adiffraction equation as θ≈mλ/d for small angles, wherein m is an orderof diffraction and d is the period of the grating. The resultingdiffracted optical flux is then reflected by the reflective surface 106to the reflective surface 104 and then to the diffraction grating 110 soas to be incident to the diffraction grating 110 at the angle θ and isthereby diffracted so as to propagate parallel to but displaced adistance Δ from the axis 108. The displaced flux is then directed by thePBS 102 to the lens 118. The counter-propagating optical flux (i.e., theflux reflected by the PBS 102) is similarly displaced a distance Δ fromthe axis 108, but in an opposite direction and is directed to the lens118 so that the counter-propagating fluxes are combined at a focal planearray detector 130 or other detector. A detected intensity distributioncan be stored in one or more computer readable media for processing byan image processor 132.

Optical path difference (OPD) associated with a focused, sheared opticalflux is illustrated in FIG. 10. As shown in FIG. 10, a shearing opticalsystem 1000 such as described above produces shear S_(shear) betweenflux portions propagating along ray directions 1002, 1003 to a lens 1006that combines the flux portions at a focal plane array (FPA) 1008 orother detector. For convenient illustration, the lens 1006 is shown as asinglet lens, but in other examples, multi-element lenses, reflectiveoptics, or catadioptric optics can be used. Referring to FIG. 10,

OPD=S _(shear) sin(θ)≈S _(shear)θ,

for small angle θ. In FIG. 10, θ is depicted as an angle in the objectspace of the lens 1006 with respect to ray directions 1002, 1003. Thisassumes that the singlet lens 1006 has an aperture stop that is locatedat the lens 1006. In this special case, θ is the angle of the chief rayin both object and image space. However, in more sophisticated lenssystems, θ is the angle of the chief ray in image space.

When the two sheared portions of the optical flux are combined by thelens, interference fringes are produced on the FPA 1008. Thisinterference can be expressed as

${{I\left( {x_{i},y_{i\;}} \right)} = {\langle{{{\frac{1}{\sqrt{2}}{E_{x}\left( {x_{i\;},y_{i},t} \right)}^{- {j\varphi}_{1}}} + {\frac{1}{\sqrt{2}}{E_{y}\left( {x_{i},y_{i},t} \right)}^{- {j\varphi}_{2}}}}}^{2}\rangle}},$

where < > represents a time average, x_(i) and y_(i) are image-planecoordinates, and φ₁, φ₂, are the cumulative phases along each ray.Expansion of this expression yields

${{I\left( {x_{i},y_{i}} \right)} = {\frac{1}{2}\begin{Bmatrix}{\left( {{\langle{E_{x}E_{x}^{*}}\rangle} + {\langle{E_{y}E_{y}^{*}}\rangle}} \right) + {\left( {{\langle{E_{x}E_{y}^{*}}\rangle} + {\langle{E_{x}^{*}E_{y}}\rangle}} \right){\cos \left( {\varphi_{1} - \varphi_{2}} \right)}} +} \\{{j\left( {{- {\langle{E_{x}E_{y}^{*}}\rangle}} + {\langle{E_{x}^{*}E_{y}}\rangle}} \right)}{\sin \left( {\varphi_{1} - \varphi_{2}} \right)}}\end{Bmatrix}}},$

where E_(x), E_(y) are now understood to be functions of image planecoordinates x_(i) and y_(i). The phase factors are

$\varphi_{1} = {{\frac{2{\pi\Delta}}{\lambda \; f_{obj}}x_{i}\mspace{14mu} {and}\mspace{14mu} \varphi_{2}} = {{- \frac{2{\pi\Delta}}{\lambda \; f_{obj}}}{x_{i}.}}}$

The Stokes parameters are defined from the components of the electricfield as

$\begin{bmatrix}S_{0} \\S_{1} \\S_{2} \\S_{3}\end{bmatrix} = {\begin{bmatrix}{{\langle{E_{x}E_{x}^{*}}\rangle} + {\langle{E_{y}E_{y}^{*}}\rangle}} \\{{\langle{E_{x}E_{x}^{*}}\rangle} - {\langle{E_{y}E_{y}^{*}}\rangle}} \\{{\langle{E_{x}E_{y}^{*}}\rangle} + {\langle{E_{x}^{*}E_{y}}\rangle}} \\{j\left( {{\langle{E_{x}E_{y}^{*}}\rangle} - {\langle{E_{x}^{*}E_{y}}\rangle}} \right)}\end{bmatrix}.}$

Re-expressing I using the definitions of the Stokes parameter and φ₁,φ₂, yields

${I\left( {x_{i},y_{i}} \right)} = {\frac{1}{2}\left\lbrack {S_{0} + {S_{2}{\cos \left( {\frac{4\pi \; \Delta}{f_{obj}}x_{i}} \right)}} - {S_{3}{\sin \left( {\frac{4\pi \; \Delta}{f_{obj}}x_{i}} \right)}}} \right\rbrack}$

Consequently, the shear modulates S₂ and S₃ onto a carrier frequency,while S₀ remains as an un-modulated component. The carrier frequency Uis a function of shear and is given by

$\begin{matrix}{U = \frac{2\pi \; {S(\lambda)}}{\lambda \; f}} & (2)\end{matrix}$

Fourier filtering can then be used to calibrate and reconstruct thespatially-dependent Stokes parameters over the image plane.

The determination of the displacement Δ as a function of interferometergeometry is illustrated in the partial unfolded layout of FIG. 1B. Thedisplacement Δ is dependent on the grating-reflective surface axialseparations b₁=b₂=b and the axial separation a of the reflectivesurfaces 104, 106. For small angles, the angular deviation θ from theon-axis path can be expressed as:

$\begin{matrix}{{\theta \approx \frac{m\; \lambda}{d} \approx \frac{\Delta}{{2b} + a}},} & (3)\end{matrix}$

wherein λ is the optical flux and m is a diffraction order. The totalshear S (λ)=2Δ can then be expressed as:

$\begin{matrix}{{S(\lambda)} = {{2\Delta} = {\frac{m\; \lambda}{d}\left( {{4b} + {2a}} \right)}}} & (4)\end{matrix}$

Thus, the generated shear is directly proportional to wavelength.

The focusing lens 118 combines the sheared optical fluxes at thedetector 130 so as to produce fringes (i.e., intensity modulation) at aspatial carrier frequency U based on the total shear, i.e., at a spatialcarrier frequency U given by:

$\begin{matrix}{{U = {\frac{2\pi \; {S(\lambda)}}{\lambda \; f} = \frac{2\pi \; {m\left( {{4b} + {2a}} \right)}}{df}}},} & (5)\end{matrix}$

wherein f is a focal length of the lens 118, and d is a grating period.

In some examples, gratings of different periods and situated to diffractat different orders are used, and the shear is given by:

${{S(\lambda)} = {{2\Delta} = {{\lambda \left( {\frac{m_{1}}{d_{1}} + \frac{m_{2}}{d_{2}}} \right)}\left( {{2b} + a} \right)}}},$

wherein m₁ and m₂ are grating diffraction orders, and d₁ and d₂ aregrating periods.

Because the shear is wavelength dependent, the spatial frequency U ofthe interference fringes which contain the polarization information fromthe scene is consequently wavelength independent in a paraxialapproximation. As a result, high visibility fringes can be obtained forbroadband optical sources, regardless of the spatial or temporalcoherence of the received optical flux. In addition, a fringe period Ucan be selected by changing one or more of the reflective surfacespacing a, grating spacings b₁, b₂, grating period d, diffraction orderm, and focal length f of the lens 118. In the example of FIG. 1B, thegrating-reflective surface spacing is the same for both the gratings110, 112, but in other examples can be different.

The example of FIGS. 1A-1B is based on a Sagnac interferometer design inwhich the two optical fluxes to be combined counter-propagate along acommon optical path. Thus, such a configuration tends to be resistant tovibration, and input optical fluxes of limited spatial and/or temporalcoherence can be used. In other examples, gratings can be situated ininterferometers of other configurations, particularly division ofamplitude interferometers so as to produce similar shear. For example,diffraction gratings can be used in conjunction with a Mach-Zehnderinterferometer to produce shear, although adequate interference fringevisibility may require appreciable optical flux coherence as the MachZehnder interferometer does not provide a common optical path.Accordingly, in applications to broad wavelength ranges, common pathinterferometers generally provide superior results.

In some applications, measurement of all four Stokes parameters isunnecessary. For example, S₃ is typically negligible in the thermalinfrared and loss of the capability of measuring circular polarization(i.e., S₃) is of little consequence. If measurement of S₃ isunnecessary, an interferometer system similar to that of FIG. 1A can beprovided with an achromatic quarter wave retarder situated with its fastaxis at 45 degrees to the axis of the PBS 102 at an interferometerinput. Such a configuration permits measurement of S₀, S₁, and S₂. Anintensity distribution I(x, y) generated at a focal plane array withsuch a system can be expressed as:

$\begin{matrix}{{I\left( {x,y} \right)} = {{\frac{1}{2}{S_{0}\left( {x,y} \right)}} - {\frac{1}{2}{{S_{12}\left( {x,y} \right)}}{\cos \left\lbrack {{2\pi \; {Uy}} - {\arg \left\{ {S_{12}\left( {x,y} \right)} \right\}}} \right\rbrack}}}} & (6)\end{matrix}$

wherein U is the shear generated by the interferometer, S₁₂=S₁+jS₂, sothat |S₁₂| is a degree of linear polarization and arg{S₁₂} is anorientation of the linear polarization.

Stokes parameters can be extracted from this intensity distribution asshown in FIG. 2. A recorded fringe intensity I(x,y) is received at 202,and at 204, the recorded intensity is Fourier transformed with respectto the shear axis (in the example of FIGS. 1A-1B, a y-axis). At 206,spatial frequency components at zero frequency and at spatial frequencyU are identified that are associated with particular combinations ofStokes parameters, such as S₀(x,y) and S₁₂=S₁+jS₂ as shown above. At208, spatial distributions of the Stokes parameters are calculated basedon the selected frequency component. Typically, the selected componentsare inverse Fourier transformed for use in estimating the associatedStokes parameter distributions.

A representative interferometer based polarimetry system configured toobtain a linear state of polarization distribution associated with ascene is illustrated in FIG. 3. As shown in FIG. 3, a modified Sagnacinterferometer 300 includes an input PBS 302, diffraction gratings 304,306 and an output linear polarizer 308. An optical flux associated witha scene is directed through an entrance aperture 310 and a quarter waveretarder 312 to the interferometer 300. An objective lens 314 issituated to produce an image that contains modulated polarizationinformation on a focal plane array 316 by combining sheared,counter-propagating optical fluxes.

Example 2 Generalized Dispersion Compensated Sagnac InterferometerSystems

A generalized Sagnac interferometer based polarimeter is illustrated inFIGS. 4A-4B. As shown in FIG. 4A, an object 402 is situated on an axis401 so that an optical flux from the object 402 is directed to acollimating lens 404 and to a PBS 406. In some examples, the collimatinglens 404 can be omitted. A portion of the optical flux in a firstpolarization state (shown as in the plane of FIG. 4A) is directedthrough a first grating 408 to mirrors 412, 414, and then to a secondgrating 410 and the PBS 406. This portion is then directed to ananalyzer 416 and focused by an objective lens 418 to a focal plane array420. A portion of the input optical flux in a second polarization state(shown in FIG. 4A as perpendicular to the plane of FIG. 4A) isoppositely directed and is combined with the counter-propagating flux inthe first polarization state at the focal plane array 420 by the lens418. The combination of the counter-propagating fluxes at the focalplane array produces an interference pattern I(x,y) that can be used todetermine one or more of the Stokes parameters or provide otherindication of polarization.

For identical diffraction gratings G₁ and G₂ with grating period d, theshear S_(DCPSI) is given by:

$\begin{matrix}{S_{DCPSI} = {\frac{2m\; \lambda}{d}\left( {a + b + c} \right)}} & (7)\end{matrix}$

wherein a, b, and c represent the distances between G₁ and M₁, M₁ andM₂, and M₂ and G₂, respectively, and m is a diffraction order. FIG. 4Billustrates the sheared optical flux in a plane 422 that isperpendicular to a z-axis. An undiffracted component of the input fluxis situated on axis at 434 while counter-propagating diffractedcomponents associated with a longer and a shorter wavelength aredisplaced to locations 430, 433 and 432, 431, respectively.

The combined output optical flux as focused by the objective lens (focallength f_(obj)) produces an intensity distribution:

$\begin{matrix}{{I_{DCPSI}\left( {x_{i},y_{i}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 0}^{d/\lambda_{\min}}{S_{0}^{\prime}(m)}}} + {\frac{1}{2}{\sum\limits_{m = 1}^{d/\lambda_{\min}}{\begin{bmatrix}{{{S_{2}^{\prime}(m)}{\cos \left( {\frac{2\pi}{f_{obj}}\frac{2m}{d}\left( {a + b + c} \right)x_{i}} \right)}} -} \\{{S_{3}^{\prime}(m)}{\sin \left( {\frac{2\pi}{f_{obj}}\frac{2m}{d}\left( {a + b + c} \right)x_{i}} \right)}}\end{bmatrix}.}}}}} & (8)\end{matrix}$

The intensity distribution I_(DCPSI) is a summation from a diffractionorder m=0 to a maximum diffraction order m=(d/λ_(min))sin(π/2), whereinλ_(min) is a shortest wavelength component of a combined optical flux atthe detector. The Stokes parameters S₀′(m), S₂′(m), and S₃′(m) asweighted by grating diffraction efficiency E(λ,m) are given by:

$\begin{matrix}{{{S_{0}^{\prime}(m)} = {\int_{\lambda_{\min}}^{\lambda_{\max}}{D\; {E^{2}\left( {\lambda,m} \right)}{S_{0}(\lambda)}{\lambda}}}},} & (9) \\{{{S_{2}^{\prime}(m)} = {\int_{\lambda_{\min}}^{\lambda_{\max}}{D\; {E^{2}\left( {\lambda,m} \right)}{S_{2}(\lambda)}{\lambda}}}},} & (10) \\{{{S_{3}^{\prime}(m)} = {\int_{\lambda_{\min}}^{\lambda_{\max}}{D\; {E^{2}\left( {\lambda,m} \right)}{S_{3}(\lambda)}{\lambda}}}},} & (11)\end{matrix}$

wherein λ_(m) in and λ_(max) are the minimum and maximum wavelengths inthe combined optical flux. Spatial carrier frequencies are given by:

$\begin{matrix}{{U_{DCPSI} = {\frac{2m}{{df}_{obj}}\left( {a + b + c} \right)}},} & (12)\end{matrix}$

which is independent of wavelength (i.e., lacks dispersion), permittingwhite-light interference fringes to be generated. In addition, carrierfrequency depends on the diffraction order m, and this dependence can beused in multispectral imaging by, for example, substitutingmultiple-order gratings for single order gratings. The diffractionefficiency weighted Stokes parameters can be obtained by demodulatingI_(DCPSI) with respect to one or more of spatial frequencies U_(DCPSI).

Example 3 White Light Polarimetric Reconstructions in S₁ and S₂

A quarter wave retarder (QWR) oriented at 45° in front of a simplifiedchanneled spectropolarimeter such as shown in FIG. 4A can be used tomeasure linear polarization (S₀, S₁, and S₂). The Mueller matrix for aQWR at 45° is

$M_{{QWR},45^{{^\circ}}} = \begin{bmatrix}1 & 0 & 0 & 0 \\0 & 0 & 0 & {- 1} \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0\end{bmatrix}$

Multiplication of this matrix by an arbitrary incident Stokes vectoryields

S _(out) =M _(QWR,45°) [S ₀ S ₁ S ₂ S ₃]^(T) =[S ₀ −S ₃ S ₂ S ₁]^(T).

Therefore, the QWR converts any incident linear horizontal or verticalpolarization states (S₁) into circular polarization (S₃) and vice versa.Consequently, with an included QWR, the detected intensity patternbecomes

${{I_{DCPSI}\left( {x_{i},y_{i}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 0}^{d/\lambda}{S_{0}^{\prime}(m)}}} + {\frac{1}{2}{\sum\limits_{m = 1}^{d/\lambda_{1}}\begin{bmatrix}{{{S_{2}^{\prime}(m)}{\cos \left( {\frac{2\pi}{f_{obj}}\frac{2m}{d}\left( {a + b + c} \right)x_{i}} \right)}} -} \\{{S_{1}^{\prime}(m)}{\sin \left( {\frac{2\pi}{f_{obj}}\frac{2m}{d}\left( {a + b + c} \right)x_{i}} \right)}}\end{bmatrix}}}}},$

wherein S₁′(m) is analogous to S₃′(m), and is defined as

S₁^(′)(m) = ∫_(λ₁)^(λ₂)D E²(λ, m)S₁(λ)λ.

Inverse Fourier transformation of channels C₀ (zero frequency component)and C₁ (component at frequency U_(DCPSI)) yields

$\begin{matrix}{{{\left\lbrack C_{0} \right\rbrack} = \frac{S_{0}^{\prime}(1)}{2}}{{{\left\lbrack C_{1} \right\rbrack} = {\frac{1}{4}\left( {{S_{2}^{\prime}(1)} + {j\; {S_{1}^{\prime}(1)}}} \right){\exp \left( {j\; 2\pi \; U_{DCPSI}x_{i}} \right)}}},}} & \mspace{11mu}\end{matrix}$

assuming that the m=1 diffraction order is dominant. Thus, a full linearpolarization measurement including the degree of linear polarization(DOLP) and its orientation can be calculated from a single interferencepattern. The DOLP and its orientation can be determined as:

${D\; O\; L\; P} = \frac{\sqrt{S_{1}^{2} + S_{2}^{2}}}{S_{0}}$$\varphi = {\frac{1}{2}{{{atan}\left( \frac{S_{2}}{S_{1}} \right)}.}}$

Example 4 Dual-Band Snapshot Imaging Polarimeter

Blazed gratings can have high diffraction efficiency into a singlediffraction order at a design wavelength. At other wavelengths, a blazedgrating can produce substantial diffraction into a plurality ofdiffraction orders. In some examples, polarization analysis can beprovided in two or more wavelength bands that are nearly integermultiples of each other. For example, analysis in a combination of amidwavelength infrared band (MWIR) of about 3-5 μm and a long wavelengthinfrared band (LWIR) of about 8-12 μm can be provided. These wavelengthbands are close to an integer separation in optical path difference sothat a blazed grating designed for a +1 order at a wavelength of 8 μmwill have maximum efficiency at 8 μm in the +1 order, 4 μm in the +2order, 2 μm in the +3 order, etc. Therefore, a grating can be chosen tobe suitable for both MWIR and LWIR bands. Diffraction efficiencies for arepresentative grating having a design wavelength of 8 μm at variousdiffraction orders is shown in FIG. 5. As shear is proportional todiffraction order, such a configuration produces twice as much shear inthe MWIR than in the LWIR so that fringe spatial frequency in the MWIRis twice that in the LWIR. MWIR and LWIR image contributions can beseparated by demodulation of the fringes based on corresponding fringespatial frequencies. Other diffraction orders can also appear in thedetected fringes, and these can be reduced or removed based on theirdiffering spatial frequencies.

Example 5 Deep Grating Multispectral Snapshot Imaging Spectrometer

As shown above, in dual-band operation, an MWIR carrier frequencygenerated by a second order diffraction order is twice that of the LWIRcarrier frequency generated by a first diffraction order. In additionalexamples, scene spatial information over a wide wavelength range can bemodulated onto carrier frequencies that are spectrally dependent so thatpolarization information or spectral information can be extracted. Insuch applications, a ‘deep’, or multiple-order blazed grating (MBG)having multiple diffraction orders spanning the wavelength region ofinterest can be used. FIG. 6 is a graph of diffraction efficiency ofsuch an MBG for a wavelength range spanning the visible and nearinfrared spectrum for diffraction orders 5-10. FIGS. 7A-7B arecross-sectional views of a single order BG 700 and an MBG 710. Both aredefined by periodic steps of triangular cross-section between refractiveindices n₁ and n₂ with period d, but the BG 700 has a height h₁ which issmaller than a height h₂ of the MBG 710.

Theoretical diffraction efficiency (DE) for an ideal blazed grating at awavelength λ in a diffraction order m can be calculated as

$\begin{matrix}{{{D\; {E\left( {\lambda,m} \right)}} = {{sinc}^{2}\left( \frac{m - {OPD}}{\lambda} \right)}},} & (13)\end{matrix}$

wherein

OPD=h(n ₁ −n ₂),  (14)

and h is groove height, OPD is an optical path difference, and n₁, n₂are indices of refraction for incident medium and blaze medium,respectively.

Example 6 Back-to-Back Grating Multispectral Snapshot ImagingSpectrometer

In other examples, multispectral polarimeters can include back-to-backgratings or grating assemblies with grating segments of various periodsand orientation. With reference to FIG. 8A, a multispectral imagingpolarimeter 800 includes an aperture 802 and a PBS 804 situated along anaxis 801 and configured to receive an input optical flux, typically anoptical flux associated with a two dimensional scene. The PBS 804 issituated to transmit a first polarization component of the input outputflux to a first multi-wavelength blazed grating (MBG) 810, mirrors 812,814, a second MBG 816 for transmission by the PBS 804 to a linearpolarizer 818. An objective lens 820 focuses the received flux onto afocal plane array detector (FPA) 824. The PBS 804 is situated to reflecta second polarization component of the input output flux to the secondMBG 816, mirrors 814, 812, the first MBG 810 for reflection by the PBS804 to the linear polarizer 818. The objective lens 820 focuses thereceived flux onto the FPA 824 in combination with the flux transmittedby the PBS 804. As a result, a fringe pattern is formed on the FPA 824,with spatial carrier frequencies proportional to diffraction order.

The MBGs 810, 812 can be deep gratings as described above and shown inFIG. 7B. Such gratings produce fringe modulations at a variety offrequencies for corresponding spectral components of the scene opticalflux based. Back to back gratings or multi-segmented gratings can beused. In the example of FIG. 8A, the MBGs 810, 812 are multi-segmentedgratings as shown in FIG. 8B. For example, the MBG 810 can comprisegrating segments 840-843 each having a different orientation and gratingperiod. The grating segments can be low order blazed gratings or MBGs aswell. The grating segments 840-843 can produce shears of differentmagnitudes and in different directions. In one example, an intensitydistribution 820 is illustrated in a plane perpendicular to a z-axis(direction of optical flux propagation) and situated between the lens820 and the analyzer 818. Shear of the input optical flux to locationsdisplaced along both the x-axis and the y-axis and combinations of suchshears is apparent.

If a linear polarizer is inserted with its axis at 45° with respect tothe x-axis, then the Stokes vector incident on the PBS 804 is given by:

$\begin{matrix}{S_{WGBS} = {{{\frac{1}{2}\begin{bmatrix}1 & 0 & 1 & 0 \\0 & 0 & 0 & 0 \\1 & 0 & 1 & 0 \\0 & 0 & 0 & 0\end{bmatrix}}\begin{bmatrix}S_{0,{inc}} \\S_{1,{inc}} \\S_{2,{inc}} \\S_{3,{inc}}\end{bmatrix}} = {\begin{bmatrix}{S_{0,{inc}} + S_{2,{inc}}} \\0 \\{S_{0,{inc}} + S_{2,{inc}}} \\0\end{bmatrix}.}}} & (15)\end{matrix}$

S₀, S_(1,inc), S_(2,inc), and S_(3,inc) are the incident Stokesparameters at the linear polarizer and are implicitly dependent uponwavelength (λ). Substituting the values from S_(WGBS) for the Stokesparameters from the equations above yields:

$\begin{matrix}{{{S_{0}^{\prime}(m)} = {{S_{2}^{\prime}(m)} = {\int_{\lambda_{\min}}^{\lambda_{\max}}{{{{DE}^{2}\left( {\lambda,m} \right)}\left\lbrack {{S_{0,{inc}}(\lambda)} + {S_{2,{inc}}(\lambda)}} \right\rbrack}{\lambda}}}}},} & (16)\end{matrix}$

Substituting the values for S₀′(m), S₂′(m), and S₃′(m) yields theintensity pattern:

$\begin{matrix}{{{I_{MSI}\left( {x_{i},y_{i}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 0}^{{Ce}{\lbrack{\lambda_{1}/\lambda_{\min}}\rbrack}}\left\lbrack {S_{0}^{''}(m)} \right\rbrack}} + {\frac{1}{2}{\sum\limits_{m = 1}^{{Ce}{\lbrack{\lambda_{1}/\lambda_{\min}}\rbrack}}\left\lbrack {{S_{0}^{''}(m)}{\cos \left( {\frac{2\pi}{f_{obj}}\frac{2m}{d}\left( {a + b + c} \right)x_{i}} \right)}} \right\rbrack}}}},{wherein}} & (17) \\{\mspace{79mu} {{S_{0}^{''}(m)} = {\int_{\lambda_{\min}}^{\lambda_{\max}}{D\; {{E^{2}\left( {\lambda,m} \right)}\left\lbrack {{S_{0,{inc}}(\lambda)} + {S_{2,{inc}}(\lambda)}} \right\rbrack}{{\lambda}.}}}}} & (18)\end{matrix}$

It should be noted that the dominant orders experimentally observed inthe system correspond to the ceiling (Ce) of λ₁/λ_(min), where λ₁ is thefirst order blaze wavelength of the diffraction grating. This changesthe maximum limit of the summation from d/λ_(min) to Ce[λ₁/λ_(min)].

Example 7 Combined Gratings/Reflectors

With reference to FIG. 9, a Sagnac interferometer based polarimeterincludes mirrors 904, 906 that include diffraction gratings 905, 907 atrespective mirror surfaces. Shear is dependent on pupil position in they-plane due to the variation in separation along the mirror local x-axesx_(l). The on-axis shear is:

$\begin{matrix}{{S(\lambda)} = \frac{2{am}\; \lambda}{d}} & (19)\end{matrix}$

wherein a is a separation between mirrors 904, 906 along an optical axis901 and is a function of x_(l). To correct or compensate, a slowlyvarying chirp can be added to the blazed gratings on the mirrors 904,906 such that a grating period d depends upon x_(l). With such amodification, shear S can be constant or nearly so over the entirepupil.

Example 8 Serial or Parallel Sagnac Interferometer Systems

In some applications, determination of all four Stokes parameter may bedesirable. Representative systems are illustrated in FIGS. 11A-11B.Referring to FIG. 11A, first and second Sagnac interferometer systems1102, 1104 that include diffraction gratings as described in theexamples above are configured to receive respective portions of an inputoptical flux 1106 from a beam splitter 1108. Typically, the beamsplitter 1108 is substantially polarization independent, and can beprovided as a plate beam splitter or other suitable optical element. TheSagnac interferometers direct sheared optical fluxes to respectivepolarizers (or other polarization components) 1110, 1112, lenses 1114,1116, and array detectors 1118, 1120, respectively. An image processor1122 receives detected interference signals from the array detectors1118, 1120, respectively, and produces estimates of some or all Stokesparameters.

FIG. 11B illustrates a representative serial configuration that permitsestimation of all four Stokes parameters. This configuration includesSagnac interferometer systems 1152, 1154 situated in series. Theinterferometer 1152 is situated to receive an input optical flux 1156and produce a sheared output flux 1158 that is directed to a retardersuch as a quarter waver retarder or half wave retarder or other retarderand directed to the interferometer 1154. The interferometer 1154provides additional shear and the sheared output is directed to ananalyzer 1160, a lens 1162, and an array detector 1164. A detectedinterference pattern is evaluated in an image processor 1166 that isconfigured to identify one or more spatial frequency components in thedetected interference pattern so as to provide estimates of one or moreStokes parameters.

The interferometers 1152, 1154 can be configured so as to produceinterference patterns at different spatial frequencies based on, forexample, diffraction grating periods, diffraction orders, or grating ormirror spacings. Modulations imposed by the interferometers can bedetected based on these differing spatial frequencies. Alternatively,the interferometers 1152, 1154 can be configured to provide modulationsat spatial frequencies associated with different spatial directions. Forexample, a first interferometer can provide an x-modulation and a secondinterferometer can provide a y-modulation that can be at the same ordifferent spatial frequency so that modulation associated with theStokes parameters can be identified based on either direction or spatialfrequency or both.

Example 9 Calcite Blazed Grating Pairs

Imaging or other polarimeters suitable for use white light or otherbroadband radiation can be based on polarization dependent diffractiongratings. Such polarimeters can produce modulated fringe patterns fromwhich one or more Stokes images can be extracted as described above withSagnac interferometer produced shear. FIG. 12 illustrates a portion of arepresentative optical system that includes a grating pair 1202 thatincludes a first grating 1204 and a second grating 1206. Forconvenience, the grating pair 1202 is described with reference to anorthogonal xyz-coordinate system 1207. A Y-axis and a Z-axis are shownin the plane of the drawing, and an X-axis is perpendicular to the planeof the drawing. The grating 1204 includes first and second birefringentsubgratings 1210, 1211 having shaped (“blazed”) surfaces 1214, 1215,respectively, that are periodic along the Y-axis. As shown in FIG. 12,the subgratings 1210, 1211 also include planar surfaces 1218, 1220 thatcan serve as optical input/output surfaces. These surfaces areatypically planar, but non-planar surfaces can be used. The surfaces1214, 1215 can be formed by any convenient process such as ruling oretching. In one convenient implementation, the surfaces 1214, 1215 areformed using anisotropic etching of a birefringent material such ascalcite. The shaped surfaces 1214, 1215 are situated so as to be spacedapart and facing each other along an axis 1220. In addition, thesurfaces 1214, 1215 are optically coupled with an index matchingmaterial 1222 such as an index matching liquid.

The birefringent subgratings 1210, 1211 can be formed of a uniaxial orbiaxial material. The shaped surface 1214 and the subgrating 1210 areconfigured so that a selected input polarization (shown in FIG. 12 as ans-polarization) propagates in the subgrating 1210 and experiences afirst index of refraction n_(o) as so-called “ordinary” ray. Arefractive index of the index matching material 1222 is selected to besubstantially equal to n_(o) so that an input beam corresponding to an“ordinary ray” is undiffracted and unrefracted at the shaped surface1214. As shown in FIG. 12, the subgrating 1210 is configured so that anX-directed linear polarization is undiffracted by the shaped surface1214 and propagates along a path 1224 that is substantially unchanged bythe shaped surface 1214. In contrast, an orthogonal polarization (aY-directed linear polarization, referred to as a p-polarization in FIG.12) is diffracted/refracted based on a refractive index differencebetween the refractive index of the matching material 1222 and anextraordinary refractive index n_(o) in the subgrating 1210 andpropagates along a path 1226. The subgrating 1211 and the shaped surface1215 are similarly configured so that the ordinary polarization exitsthe grating 1204 along the path 1226 and the extraordinary polarizationis diffracted/refracted by the shaped surface 1215 so as to exit thegrating 1204 along a path 1230 that is substantially parallel to andoffset from the path 1226 associated with the ordinary ray.

The displaced ordinary and extraordinary beams could be combined with alens and at least partially projected into a common state ofpolarization with a polarizer that is unaligned with either to produceinterference fringes. Unfortunately, the displaced beams are associatedwith significant phase delays so that broadband illumination wouldproduce no fringes or fringes with limited visibility. To compensate,the second grating 1206 is configured similarly to the first grating1204, but with birefringent subgratings 1240, 1241 arranged so that theordinary beam is diffracted/refracted along path 1232 by a shapedsurface 1244 and then along a path 1234 by a shaped surface 1245. Thus,the same diffraction angles are encountered by both polarizationcomponents, although in different polarizing gratings. As a result,orthogonally polarized beams exit the second grating 1206 alongparallel, displaced paths 1230, 1234. With each polarization displaced,broadband or white light fringes can be obtained. The shear S betweenthe paths 1230, 1234 can be obtained as S=2 d_(g) tan θ≈2 d_(g)λ/T,wherein λ is a wavelength, θ is a diffraction angle, m is a diffractionorder, T is a grating period, and d_(g) is separation of the shapedsurfaces 1214, 1215 or 1244, 1245. As discussed above, a shear that isproportional to wavelength results in a spatial carrier frequency thatis independent of wavelength, and thus suitable for use with broadbandradiation.

In the example of FIG. 12, grating periods and shaped surfaceseparations are the same for both the first and second gratings 1204,1206, but in other examples, different spacing and periods can be used.Typically, differing periods and/or spacings tend to produce lessvisible fringes with broadband radiation. The configuration of FIG. 12produces a compensated shear suitable for use with broadband radiation.

Example 10 Dual Calcite Blazed Grating Pairs for X- and Y-Displacements

Referring to FIG. 13, a first grating pair 1300 includes first andsecond calcite blazed gratings 1302, 1304 such as those described aboveand configured to produce compensated shear along a Y-direction so thatan s-polarized beam 1306 and a p-polarized beam 1308 have a compensatedshear S_(Y) along a Y-direction. One or both of the beams 1306, 1308 canbe sheared in an X-direction with a second grating pair 1320 thatincludes first and second calcite blazed gratings 1322, 1324 orientedorthogonal to those of the first grating pair 1300. The second gratingpair 1320 can be configured to produce a shear S_(X) that is the same ordifferent that the shear S_(Y) based on grating periods or spacings.

A half-waveplate (HWP)1330 is situated with a fast or slow axis at about22.5 degrees with respect to the X-axis or the Y-axis and between thefirst grating pair 1300 and the second grating pair 1320. The HWP 1330rotates the plane polarization of each of the beams 1306, 1308 by 45degrees so each of the beams 1306, 1308 is further sheared in anX-direction by the second grating pair 1320. Thus, four sheared beamsare produced. To produce interference fringes, a polarization analyzer1340 is situated so as to transmit linear polarization along an axis at45 degrees with respect to the X-axis or the Y-axis. The correspondingpolarized beams can then be focused to produce interference fringes.

Example 11 Dual Calcite Blazed Grating Pairs for Linear SOP Imaging

FIG. 14 is a representative example of a birefringent grating pair 1400configured for linear polarization measurements. A quarter waveplate1402 and first and second gratings 1406 and 1408 are situated along anaxis 1410. An analyzer 1414 is situated with a fast axis at 45 degreeswith respect to the states of polarization of the sheared beams 1410,1412. Measurements of Stokes parameters S₁ and S₂ can be obtained, andthe assembly can be located in a focal plane of a 4f imaging system(i.e., an imaging system with object and image distances of twice afocal length) or in front of a single lens/FPA combination for imagingof distant objects.

Example 12 Liquid Crystal Polarization Grating (PG) Pairs

With reference to FIG. 15, a first liquid crystal grating (LCG1) 1502and a second liquid crystal grating (LCG2) 1504 are situated on an axis1506 so as to receive an input optical flux 1508 and produce a shear Sbetween first and second polarization components. In the example of FIG.15, the LCG 1502 diffracts an s-component upward into a +1 diffractionorder along 1508 and a p-component downward into a −1 diffraction orderalong 1510. The LCG 1504 is also situated to diffract the s-componentreceived from the LCG 1502 into a +1 diffraction order so as to exit theLCG 1504 along an axis 1512 that parallel to and displaced upwardly fromthe axis 1506. In addition, the LCG 1504 diffracts the receivedp-component into a −1 diffraction order along an axis 1514 that isparallel to and displaced downwardly from the axis 1506.

For convenience, FIG. 15 is described with reference to particularorthogonal linear polarizations, but any orthogonal polarization statescan be similarly processed using one or more quarter waveplates, halfwaveplates, or other retardation plates, typically situated prior to theLCG 1502.

Example 13 Channeled Imaging Polarimeter Using PGs

FIG. 16 illustrates a representative channeled imaging polarimeter (CIP)1600. The CIP 1600 is configured to interferometrically generate carrierfrequencies that are amplitude modulated based on spatially-dependent2-dimensional Stokes parameters. Such a CIP exhibits inherent imageregistration and can be implemented with simple optical components.Image registration is inherent as all the Stokes parameters are directlymodulated onto coincident interference fringes and the shear producingoptical components can be added to nearly any pre-existing lens andcamera system.

The CLI 1600 includes a first polarizing grating (PG) 1602 and a secondPG 1604 that are situated along an axis 1606 and spaced apart by adistance t. The PGs 1602, 1604 provide shear similar to that produced bya diffractive Savart plate, so that interference fringes similar to aSagnac interferometer's white-light fringes can be produced. Some or allStokes parameters can be obtained. As shown in FIG. 16, the PGs 1602,1604 have grating period Λ, and the PG 1604 is followed by a linearpolarizer (LP) 1610 oriented with its transmission axis at 0° withrespect to an X-axis axis. An objective lens 1612 with focal lengthfimages collimated light from the PGs 1602, 1604 to produce polarizationmodulated fringes at a focal plane array (FPA) 1614.

Various PGs can be used. In a convenient example, spatially-periodicbirefringence devices are used based on liquid crystal (LC) materialssuch as described in Oh and Escuti, “Numerical analysis of polarizationgratings using the finite-difference time-domain method,” Phys Rev A 76(4), 043815 (2007), Oh and Escuti, “Achromatic diffraction frompolarization gratings with high efficiency,” Opt. Lett. 33, 2287-2289(2008), Crawford et al., “Liquid-crystal diffraction gratings usingpolarization holography alignment techniques,” J Appl Phys 98, 123102(2005), Escuti et al., “Simplified spectropolarimetry using reactivemesogen polarization gratings,” Proc. SPIE 6302, 630207, (2006), Escutiet al., U.S. Patent Application Publication 2010/0110363, and Escuti etal., U.S. Patent Application Publication 2010/0225856, all of which areincorporated herein by reference.

Such PGs can serve as thin-film beamsplitters that are functionallyanalogous to Wollaston prisms. In both elements, incident light isangularly separated into two, forward-propagating, orthogonalpolarizations. However, typical PGs are an embodiment of thePancharatnam-Berry phase operating on circular eigen-polarizations,whereas Wollaston prisms are based on double refraction and operate onlinear eigen-polarizations. Details of LC microstructure and holographicfabrication can be found in the references noted above.

The polarization behavior and diffraction efficiency spectra of such LCPGs are different than conventional phase or amplitude gratings. Whilethe natural eigen-polarizations are circular (i.e., linearlyproportional to S₃/S₀), LC PGs can be paired with a quarter waveplate(QWP) in order to separate incident light based on other desiredpolarizations (i.e., S₁/S₀ or S₂/S₀). Light diffracted from the PGs isdirected almost entirely into the first (m=±1) or zero (m=0) diffractionorders, wherein diffraction angles are defined by the classical gratingequation sin θ_(m)=mλ/Λ−sin θ_(in), wherein Λ is the grating period, mis the grating order, and θ_(m) and θ_(in), are the diffracted andincidence angles, respectively. The diffraction efficiency of a PG canbe typically expressed as:

${\eta_{\pm 1} = {\left( {\frac{1}{2} \mp \frac{S_{3}}{2S_{0}}} \right)K}},{\eta_{0} = \left( {1 - K} \right)},$

wherein K is a factor determined by the LC structure in the PG.

The CLI polarimeter 1600 preferably uses PGs that are capable of highefficiency operation over a broad (white-light) spectrum. The originalLC-based PG had a relatively narrow diffraction efficiency spectrum suchthat high first-order efficiency (>99%) occurred only at wavelengthsclose to a specified design wavelength λ₀ (typically within Δλ/λ₀˜13%).However, broadband PGs having a high efficiency spectral bandwidth(Δλ/λ₀˜56%) which can cover most of the visible wavelength range areavailable. For these PGs, the factor K can be approximated as K=1, sothat η_(±1)=1 and η₀=0 for most visible wavelengths (e.g., 450-750 nm).

In the CLI polarimeter 1600, incident light is transmitted by PG₁ anddiffracted into left and right circularly polarized components,propagating above and below the axis 1606, respectively. Aftertransmission through PG₂, the two beams (E_(A) and E_(B)) are diffractedagain to propagate parallel to the optical axis 1606 and are now shearedby a distance 2α. The linear polarizer (LP) 1610 analyzes both beams,thus producing a common polarization state. Imaging both beams onto theFPA 1614 with the lens 1612 combines the two beams and producesinterference fringes.

The intensity pattern on the FPA 1614 can be estimated by assuming thatan arbitrarily polarized electric field is incident on the firstpolarization grating (PG₁). The incident field can be expressed as

${E_{inc} = {\begin{bmatrix}{\overset{\_}{E}}_{X} \\{\overset{\_}{E}}_{Y}\end{bmatrix} = \begin{bmatrix}{{E_{X}\left( {\xi,\eta} \right)}^{j\; {\phi_{x}{({\xi,\eta})}}}} \\{{E_{Y}\left( {\xi,\eta} \right)}^{j\; {\phi_{y}{({\xi,\eta})}}}}\end{bmatrix}}},$

wherein ξ, η are the angular spectrum components of x and y,respectively. The PG's +1 and −1 diffraction orders can be modeled asright and left circular polarization analyzers with their Jones matricesexpressed as

${J_{{+ 1},{RC}} = {\frac{1}{2}\begin{bmatrix}1 &  \\{- } & 1\end{bmatrix}}},{J_{{- 1},{LC}} = {{\frac{1}{2}\begin{bmatrix}1 & {- } \\ & 1\end{bmatrix}}.}}$

After transmission through PG₁ and PG₂, the x and y polarizationcomponents of the electric field, for each of the two beams, are

${E_{A} = {{J_{{- 1},{LC}}E_{inc}} = {\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)} - {j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}} \\{{j\; {{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)}} + {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}\end{bmatrix}}}},{E_{B} = {{J_{{+ 1},{RC}}E_{inc}} = {\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)} + {j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}}} \\{{{- j}\; {{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)}} + {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}}\end{bmatrix}}}},$

wherein a is the shear, calculated using the paraxial approximation as

$\alpha \cong {\frac{m\; \lambda}{\Lambda}t}$

wherein m is a diffraction order (usually either 1 or −1). The totalelectric field incident on the linear polarizer (LP) 1610 is

$E_{LP}^{+} = {{E_{A} + E_{B}} = {{\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)} + {j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}} + {{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)} -} \\{{j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}} - {j\; {{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)}} + {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)} +} \\{{j\; {{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)}} + {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}\end{bmatrix}}.}}$

Transmission through the linear polarizer, with its transmission axis at0°, yields

$E_{LP}^{-} = {{\begin{bmatrix}1 & 0 \\0 & 0\end{bmatrix}E_{LP}^{+}} = {\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)} + {j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}} + {{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)} - {j\; {{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}} \\0\end{bmatrix}}}$

The objective lens 1614 produces a Fourier transformation of the fieldas,

${E_{f} = {{F\left\lbrack E_{LP}^{-} \right\rbrack}_{{\xi = \frac{x}{\lambda \; f}},{\eta = \frac{y}{\lambda \; f}}} = {\frac{1}{2}\left\lbrack {{{\overset{\_}{E}}_{X}^{j\frac{2\pi}{\lambda \; f}\alpha \; y}} + {j{\overset{\_}{E}}_{Y}^{j\frac{2\pi}{\lambda \; f}\alpha \; y}} + {{\overset{\_}{E}}_{X}^{{- j}\frac{2\pi}{\lambda \; f}\alpha \; y}} - {j{\overset{\_}{E}}_{Y}^{{- j}\frac{2\pi}{\lambda \; f}\alpha \; y}}} \right\rbrack}}},$

wherein Ē_(X) and Ē_(Y), are now implicitly dependent upon x and y and fis the focal length of the objective lens 1614. Total electric fieldintensity can be written as follows:

$I = {{E_{f}}^{2} = {{\frac{1}{2}\left( {{{\overset{\_}{E}}_{X}}^{2} + {{\overset{\_}{E}}_{Y}}^{2}} \right)} + {\frac{1}{4}\left( {{{\overset{\_}{E}}_{X}{\overset{\_}{E}}_{X}^{*}} - {{\overset{\_}{E}}_{Y}{\overset{\_}{E}}_{Y}^{*}}} \right)^{j\frac{2\pi}{\lambda \; f}2\alpha \; y}} + {\frac{1}{4}\left( {{{\overset{\_}{E}}_{X}{\overset{\_}{E}}_{X}^{*}} - {{\overset{\_}{E}}_{Y}{\overset{\_}{E}}_{Y}^{*}}} \right)^{{- j}\frac{2\pi}{\lambda \; f}2\alpha \; y}} + {j\; \frac{1}{4}\left( {{{\overset{\_}{E}}_{X}{\overset{\_}{E}}_{Y}^{*}} + {{\overset{\_}{E}}_{Y}{\overset{\_}{E}}_{X}^{*}}} \right)^{j\frac{2\pi}{\lambda \; f}2\alpha \; y}} - {j\frac{1}{4}\left( {{{\overset{\_}{E}}_{X}{\overset{\_}{E}}_{Y}^{*}} + {{\overset{\_}{E}}_{Y}{\overset{\_}{E}}_{X}^{*}}} \right){^{{- j}\frac{2\pi}{\lambda \; f}2\alpha \; y}.}}}}$

Simplification using the Stokes parameter definitions yields the finalexpression for the intensity pattern:

$\begin{matrix}{{I\left( {x,y} \right)} = {{\frac{1}{2}\left\lbrack {{S_{0}\left( {x,y} \right)} + {{S_{1}\left( {x,y} \right)}{\cos \left( {\frac{2\pi}{\lambda \; f}2\alpha \; y} \right)}} + {{S_{2}\left( {x,y} \right)}{\sin \left( {\frac{2\pi}{\lambda \; f}2\; \alpha \; y} \right)}}} \right\rbrack}.}} & (20)\end{matrix}$

Consequently, the intensity recorded on the FPA 1614 contains theamplitude modulated Stokes parameters S₀, S₁ and S₂. Substitution of theshear into Eq. (20) produces an expression for intensity I:

$\begin{matrix}{{I\left( {x,y} \right)} = {{\frac{1}{2}\left\lbrack {{S_{0}\left( {x,y} \right)} + {{S_{1}\left( {x,y} \right)}{\cos \left( {2\pi \frac{2\; {mt}}{f\; \Lambda}y} \right)}} + {{S_{2}\left( {x,y} \right)}{\sin \left( {2\pi \frac{2{mt}}{f\; \Lambda}y} \right)}}} \right\rbrack}.}} & (21)\end{matrix}$

From Eq. (2), the frequency of the interference fringes, or the carrierfrequency, denoted by U is

$\begin{matrix}{U = {\frac{2{mt}}{f\; \Lambda}.}} & (22)\end{matrix}$

Thus, the linear Stokes parameters are amplitude modulated ontospectrally broadband (white-light) interference fringes.

Example 14 CLI Calibration

A CLI polarimeter such as that of FIG. 16 can be calibrated by applyinga reference beam calibration technique as described in Oka and Saito,“Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE6295, 629508 (2006) and Kudenov et al., “Prismatic imaging polarimetercalibration for the infrared spectral region,” Opt. Exp. 16, 13720-13737(2008), both of which are incorporated herein by reference. First, aforward 2-dimensional (2D) Fourier transformation is performed on theintensity pattern of Eq. (21), producing

$\begin{matrix}{{{I\left( {\xi,\eta} \right)} = {{F\left\lbrack {I\left( {x,y} \right)} \right\rbrack} = {{\frac{1}{2}{S_{0}\left( {\xi,\eta} \right)}} + {\frac{1}{4}{S_{1}\left( {\xi,\eta} \right)}*\left\lbrack {{\delta \left( {\xi,{\eta + U}} \right)} + {\delta \left( {\xi,{\eta - U}} \right)}} \right\rbrack} + {\frac{1}{4}{S_{2}\left( {\xi,\eta} \right)}*\left\lbrack {{\delta \left( {\xi,{\eta + U}} \right)} - {\delta \left( {\xi,{\eta - U}} \right)}} \right\rbrack}}}},} & (23)\end{matrix}$

wherein ξ and η are the Fourier transform variables for x and y,respectively, and ξ is the Dirac delta function. Eq. (23) indicates thepresence of three “channels” in the Fourier domain. The S₁ and S₂ Stokesparameters are modulated (i.e., convolved) by two shifted (±U) deltafunctions, while the S₀ Stokes parameter remains unmodulated. Thesethree channels are denoted as C₀ (S₀), C₁ ((S₁−iS₂) δ(ξ,η−U)) andC₁*((S₁+iS₂)δ(ξ,η+U)), respectively. Applying a 2D filter to two of thethree channels (C₀ and C₁ or C₁*), followed by an inverse Fouriertransformation, enables their content to be isolated from the othercomponents. Inverse Fourier transformation of channels C₀ and C₁produces

$\begin{matrix}{{C_{0} = {\frac{1}{2}{S_{0}\left( {x,y} \right)}}},} & (24) \\{C_{1} = {\frac{1}{4}\left( {{S_{1}\left( {x,y} \right)} - {\; {S_{2}\left( {x,y} \right)}}} \right){^{\; 2\; \pi \; {Uy}}.}}} & (25)\end{matrix}$

Therefore, the S₀ Stokes parameter can be extracted directly from Eq.(24), while the S₁ and S₂ components are modulated by an exponentialphase factor e^(i2πUy). Isolating this phase factor from the sample data(C_(0,sample) and C_(1,sample)) is accomplished by comparing it to apreviously measured reference polarization state (C_(0,ref) andC_(1,ref)) containing the known distribution [S_(0,ref), S_(1,ref),S_(2,ref), S_(3,ref)]^(T). The sample's Stokes parameters aredemodulated by dividing the sample data by the reference data, followedby normalization to the S₀ Stokes parameter and extraction of the realand imaginary parts,

$\begin{matrix}{\mspace{79mu} {{{S_{0}\left( {x,y} \right)} = {C_{0,{sample}}}},}} & (26) \\{{\frac{S_{1}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)} = {\left\lbrack {\frac{C_{1,{sample}}}{C_{1,{reference}}}\frac{C_{0,{reference}}}{C_{0,{sample}}}\left( \frac{{S_{1,{ref}}\left( {x,y} \right)} - {{iS}_{3,{ref}}\left( {x,y} \right)}}{S_{0,{ref}}\left( {x,y} \right)} \right)} \right\rbrack}},} & (27) \\{\frac{S_{2}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)} = {{\left\lbrack {\frac{C_{1,{sample}}}{C_{1,{reference}}}\frac{C_{0,{reference}}}{C_{0,{sample}}}\left( \frac{{S_{1,{ref}}\left( {x,y} \right)} - {{iS}_{2,{ref}}\left( {x,y} \right)}}{S_{0,{ref}}\left( {x,y} \right)} \right)} \right\rbrack}.}} & (28)\end{matrix}$

For instance, using reference data created by a linear polarizer,oriented at 0° [S₀, S₁, S₂, S₃]^(T)=[1, 1, 0, 0]^(T), yields thefollowing reference-beam calibration equations:

$\begin{matrix}{{{S_{0}\left( {x,y} \right)} = {C_{0,{sample}}}},} & (29) \\{{\frac{S_{1}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)} = {\left\lbrack {\frac{C_{1,{sample}}}{C_{1,{reference}}}\frac{C_{0,{reference}}}{C_{0,{sample}}}} \right\rbrack}},} & (30) \\{\frac{S_{2}\left( {x,y} \right)}{S_{0}\left( {x,y} \right)} = {{\left\lbrack {\frac{C_{1,{sample}}}{C_{1,{reference}}}\frac{C_{0,{reference}}}{C_{0,{sample}}}} \right\rbrack}.}} & (31)\end{matrix}$

Eqns. (29)-(31) are applied to the measured data in order to extract thescene's spatially-dependent Stokes parameters.

Example 15 CLI Polarimeter Implementation

An experimental configuration for establishing the measurement accuracyof a CLI polarimeter 1700 in white-light is illustrated in FIG. 17. Alinear polarization generator (LPG) 1704 includes a tungsten halogenfiber-lamp 1706 configured to illuminate a diffuse white ceramic plate1708. The diffuser 1708 is positioned near the focal point of acollimating lens 1710 with an effective focal length, f_(c), of 40 mm.Collimated light propagates to a linear polarizer 1712 oriented with itstransmission axis at θ. The polarization generator 1704 produces auniformly polarized scene for the CLI polarimeter 1700 to image. Theperiod of polarization gratings (PG₁ and PG₂) 1718, 1720 is Λ=7.9 m andthe focal length of an objective lens 1724 is f_(o)=23 mm. An infraredblocking filter (IRB) 1726 is situated in front of the objective lens1724 to limit the spectral passband of the imaged light to 410-750 nm.Lastly, a FPA 1730 is an 8-bit monochrome machine vision camera,containing 640×480 pixels, placed at the focal point of the objectivelens 1724. A linear polarizer 1730 is configured to analyze the shearedbeams.

Example 16 Calibration Verification

In order to verify the mathematical relationship of Eq. (2), in additionto the calibration accuracy defined at a constant illuminationcondition, reference data were taken with the apparatus of FIG. 17 androtating the LP 1712 for angles θ between 0° and 180° in 10° increments.After reconstruction, a central portion of the field of view (FOV) wasaveraged over a 100×100 pixel area to obtain an average value for themeasured polarization state. Images of the white-light interferencefringes from this 100×100 pixel area are depicted in FIGS. 18A-18C for θequal to 0°, 50° and 90°, respectively.

Note that the phase of the sinusoidal fringes changes while theamplitude remains constant for varying linear polarizer orientations.This phase change is directly related to Eq. (21), and indicates thevarying proportions of S₁ to S₂ as the LP 1712 is rotated. Meanwhile,the amplitude remains constant because the degree of linear polarization(DOLP=√{square root over (S₁ ²+S₂ ²)}/S₀) from the LPG 1704 is constant(˜1). Plotting the measured S₁ and S₂ Stokes parameters versus θ andcomparing them to the theoretical values yields the results depicted inFIG. 19. The calculated RMS error for both curves is approximately 1.6%.This implies that the amplitude modulation of Eq. (21) accuratelyfollows the incident Stokes parameter variation.

Example 17 Polarization Grating Performance

To assess the performance of PGs, zeroth diffraction order transmissionscan be measured. This provides an approximate metric for how efficientlyPGs diffract light into the +/−1 diffraction orders. Results of typicaltransmission measurements are shown in FIG. 20, demonstrating that thePGs are highly efficient for wavelengths spanning 500-750 nm, but ratherinefficient below 475 nm. Consequently, zero-order light transmitted atwavelengths less than 475 nm can cause error in the calculated Stokesparameters, primarily in the normalization of the measured Stokesparameters to S₀. Expressing the Stokes parameters in Eq. (21) asspectrally band-integrated functions yields

S_(n)^(′)(x, y) = ∫_(λ₁)^(λ₂)DE²(λ)S_(n)(x, y, λ)λ,

wherein DE is the diffraction efficiency of one PG for the +1st or −1storder, the prime superscript on the Stokes parameters indicate that theyhave been spectrally band-integrated, and the subscript n=0, 1, or 2indicates the S₀, S₁, or S₂ Stokes parameter, respectively. It isassumed for this example that both PGs have the same DE as a function ofwavelength. In a spectral region where the DE is not ideal, such thatDE<1.0, then some light transmitted through the PGs is not diffracted.This can be introduced to the model [Eq. (21)] as an additionalunmodulated zero-order undiffracted offset term Δ_(offset):

${{I\left( {x,y} \right)} = {\frac{1}{2}\left\lbrack {{\Delta_{offset}\left( {x,y} \right)} + {S_{0}^{\prime}\left( {x,y} \right)} + {{S_{1}^{\prime}\left( {x,y} \right)}{\cos \left( {2\pi \frac{2{mt}}{f\; \Lambda}y} \right)}} + {{S_{2}^{\prime}\left( {x,y} \right)}{\sin \left( {2\pi \frac{2{mt}}{f\; \Lambda}y} \right)}}} \right\rbrack}},$

Reconstructing via Eqns. (26)-(28) yields the appropriate absoluteresults for S₁ and S₂; however, S₀ will be erroneous due to theadditional offset. Therefore, measured normalized Stokes parameters canbe introduced and denoted by double primes

${{S_{0}^{''}\left( {x,y} \right)} = {{S_{0}^{\prime}\left( {x,y} \right)} + {\Delta_{offset}\left( {x,y} \right)}}},{\frac{S_{n}^{''}\left( {x,y} \right)}{S_{0}^{''}\left( {x,y} \right)} = \frac{S_{n}^{\prime}\left( {x,y} \right)}{{S_{0}^{\prime}\left( {x,y} \right)} + {\Delta_{offset}\left( {x,y} \right)}}},$

wherein the subscript n=1 or 2 indicates the S₁ or S₂ Stokes parameter,respectively. Consequently, error is induced into the S₁ and S₂ Stokesparameters from the normalization to the effectively larger S₀ component(S₀′((x, y)+Δ_(offset) (x, y)). While error due to this zero-order lightleakage was observed in some outdoor tests, it was negligible inlaboratory characterizations in which an S₀ reference and sampleillumination levels were constant. PG's with a zero-order lighttransmission less than 3% over the passband would enable better accuracyregardless of the S₀ illumination level.

Example 18 Outdoor Measurements

The snapshot imaging capability of a CLI polarimeter was also assessedoutdoors on moving targets. For outdoor scenes, the absolute accuracy ofthe Stokes parameters for varying illumination levels is not wellestablished, again due to the zero-order diffraction efficiency leakagediscussed previously. Outdoor results are provided here to demonstratesnapshot imaging and reconstruction capabilities in full sunlight.

The optical configuration for these tests is depicted in FIG. 21. A 1:1afocal telescope 2102 includes two 50 mm focal length lenses 2104, 2106operating at a focal number of F/1.8. These optics enable defocus to beintroduced into a scene image while simultaneously maintaining focus onthe fringes that are localized at infinity. Defocus is used toband-limit the spatial frequency content of the scene, thereby reducingaliasing artifacts in the reconstructed Stokes parameters. A raw imageof a moving vehicle, captured with the CLI polarimeter, is depicted inFIG. 22. The image was taken on a clear and sunny afternoon with anexposure time of approximately 1/1200 second with a re-imaging lensfocal number of F/2.5. Reference data, taken of a linear polarizeroriented at 0° in front of a diffuser, was measured shortly after thevehicle was imaged. The diffuser was illuminated by sunlight.

The polarization data was extracted by taking a fast Fouriertransformation of the raw data, followed by filtration, an inverseFourier transformation, and calibration by application of Eqns.(26)-(28). The reconstructed data were also processed with an aliasingreduction filter that reduces noise due to aliasing artifacts. Thisproduced the data depicted in FIG. 23A-23D corresponding to S₀, degreeof linear polarization (DOLP), S₁/S₀, and S₂/S₀, respectively, wherein

${{DOLP}\left( {x,y} \right)} = {\frac{\sqrt{{S_{1}^{2}\left( {x,y} \right)} + {S_{2}^{2}\left( {x,y} \right)}}}{S_{0}\left( {x,y} \right)}.}$

The orientation of the linearly polarized light (θ_(L)) can be extractedfrom the measured Stokes parameters using the formula

$\begin{matrix}{{\theta_{L}\left( {x,y} \right)} = {\frac{1}{2}{{\tan^{- 1}\left( \frac{S_{2}\left( {x,y} \right)}{S_{1}\left( {x,y} \right)} \right)}.}}} & (32)\end{matrix}$

By incorporating a color fusion method, this orientation information canbe superimposed onto the DOLP and intensity (S₀) information. In colorfusion, a hue (pixel color), saturation (amount of color within thepixel) and value (pixel brightness) color-mapping is used. ThisHue-Saturation-Value (HSV) color map is mapped directly into linearpolarization orientation (hue), DOLP (saturation), and intensity S₀(value). Images generated with this scheme provide a qualitativeassessment of polarimetric and intensity information. A color fusionimage can be generated from the image data associated with FIGS.23A-23D, along with orientation information calculated from Eq. (32)above.

Example 19 Full Stokes Polarimetry

A CLI polarimeter can be analyzed as a subset of a Savart-plate Stokesimaging polarimeter. By replacing each Savart plate with two PGs, awhite-light Stokes imaging polarimeter capable of measuring S₀, S₁, S₂and S₃ can be realized. An optical layout for this scheme is depictedschematically in FIG. 24A-24B. Light transmitted by PG₁ 2402 and PG₂2404 is sheared along a y axis by a distance α to produce two circularlypolarized diffracted beams 2407, 2408 that are converted into linearlypolarized light after propagation through a quarter wave plate (QWP)2406. Transmission through PG₃ 2410 and PG₄ 2412 shears each of thebeams 2407, 2408 along the x axis by a distance β. Propagation of thefour circularly polarized beams through a QWP 2420, linear polarizer2422, and objective lens 2424 generates white-light polarizationinterference fringes at an FPA 2426. Propagation of a single polarizedray is depicted in the perspective view in FIG. 24B.

For the purposes of the following derivation, the PG₁ to PG₂ separation(t₁) is equal to the PG₃ to PG₄ separation (t₂), such that t₁=t₂=t.Furthermore, all four PGs have an identical grating period Λ. Theincident arbitrarily polarized electric field is defined as

$E_{inc} = {\begin{bmatrix}{\overset{\_}{E}}_{X} \\{\overset{\_}{E}}_{Y}\end{bmatrix} = {\begin{bmatrix}{{E_{X}\left( {\xi,\eta} \right)}^{{j\phi}_{x}{({\xi,\eta})}}} \\{{E_{Y}\left( {\xi,\eta} \right)}^{{j\phi}_{y}{({\xi,\eta})}}}\end{bmatrix}.}}$

After transmission through PG₁ and PG₂, the x and y components of theelectric field for E_(A) and E_(B) are identical to those of Example 13above. Propagation through the QWP 2420, oriented with its fast-axis at0°, yields

$\begin{matrix}{E_{A}^{\prime} = {\begin{bmatrix}1 & 0 \\0 & {- j}\end{bmatrix}E_{A}}} \\{{= {\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)} - {j{{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}} \\{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta - \alpha}} \right)} - {j{{\overset{\_}{E}}_{Y}\left( {\xi,{\eta - \alpha}} \right)}}}\end{bmatrix}}},}\end{matrix}$ $\begin{matrix}{E_{B}^{\prime} = {\begin{bmatrix}1 & 0 \\0 & {- j}\end{bmatrix}E_{B}}} \\{= {{\frac{1}{2}\begin{bmatrix}{{{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)} + {j{{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}}} \\{{- {{\overset{\_}{E}}_{X}\left( {\xi,{\eta + \alpha}} \right)}} - {j{{\overset{\_}{E}}_{Y}\left( {\xi,{\eta + \alpha}} \right)}}}\end{bmatrix}}.}}\end{matrix}$

Propagation of E_(A)′ and E_(B)′ through PG₃ and PG₄ yields 4 beams,labeled E_(C), E_(D), E_(E) and E_(F) in FIG. 24B. These fourtransmitted fields are expressed by

$\begin{matrix}{{E_{C}\left( {{\xi + \alpha},{\eta - \alpha}} \right)} = {J_{{- 1},{LC}}{E_{A}^{\prime}\left( {\xi,{\eta - \alpha}} \right)}}} \\{{= {\frac{1}{4}\begin{bmatrix}{\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right) - {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}} \\{\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right) - {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}}\end{bmatrix}}},}\end{matrix}$ $\begin{matrix}{{E_{D}\left( {{\xi + \alpha},{\eta - \alpha}} \right)} = {J_{{+ 1},{RC}}{E_{A}^{\prime}\left( {\xi,{\eta - \alpha}} \right)}}} \\{{= {\frac{1}{4}\begin{bmatrix}{\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right) + {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}} \\{\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right) - {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}}\end{bmatrix}}},}\end{matrix}$ $\begin{matrix}{{E_{E}\left( {{\xi + \alpha},{\eta + \alpha}} \right)} = {J_{{- 1},{LC}}{E_{B}^{\prime}\left( {\xi,{\eta + \alpha}} \right)}}} \\{{= {\frac{1}{4}\begin{bmatrix}{\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right) + {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}} \\{{- \left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)} + {j\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right)}}\end{bmatrix}}},}\end{matrix}$ $\begin{matrix}{{E_{F}\left( {{\xi - \alpha},{\eta + \alpha}} \right)} = {J_{{+ 1},{RC}}{E_{B}^{\prime}\left( {\xi,{\eta + \alpha}} \right)}}} \\{{= {\frac{1}{4}\begin{bmatrix}{\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right) - {j\left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)}} \\{{- \left( {{\overset{\_}{E}}_{X} + {\overset{\_}{E}}_{Y}} \right)} - {j\left( {{\overset{\_}{E}}_{X} - {\overset{\_}{E}}_{Y}} \right)}}\end{bmatrix}}},}\end{matrix}$

where Ē_(X) and Ē_(Y) are implicitly dependent on ξ, η, and α.Transmission through the last QWP 2420, with its fast-axis oriented at45°, rotates the circular polarization states of E_(C), E_(D), E_(E) andE_(F) into vertical and horizontal linear polarizations. Propagation ofthese beams through the analyzing linear polarizer 2422 unifies theminto a 45° linear polarization state. The complete x and y components ofthe electric field incident on the lens 2424 are:

$\begin{matrix}{E_{X}^{L} = E_{Y}^{L}} \\{= {\frac{1}{4}\left( {{{\overset{\_}{E}}_{X}\left( {{\xi + \alpha},{\eta - \alpha}} \right)} - {j{{\overset{\_}{E}}_{Y}\left( {{\xi + \alpha},{\eta - \alpha}} \right)}} +} \right.}} \\{{\left( {{{\overset{\_}{E}}_{X}\left( {{\xi - \alpha},{\eta - \alpha}} \right)} - {j{{\overset{\_}{E}}_{Y}\left( {{\xi - \alpha},{\eta - \alpha}} \right)}}} \right) +}} \\{{\left( {{j{{\overset{\_}{E}}_{X}\left( {{\xi + \alpha},{\eta + \alpha}} \right)}} - {{\overset{\_}{E}}_{Y}\left( {{\xi + \alpha},{\eta + \alpha}} \right)}} \right) +}} \\{\left( {{{- j}{{\overset{\_}{E}}_{X}\left( {{\xi - \alpha},{\eta + \alpha}} \right)}} + {\left( {{\overset{\_}{E}}_{Y}\left( {{\xi - \alpha},{\eta + \alpha}} \right)} \right).}} \right.}\end{matrix}$

The lens 2424 produces a Fourier transformation of the field. Performingthis on the E_(X) ^(L) component yields

$\begin{matrix}{E_{L} = {F\left\lbrack E_{X}^{L} \right\rbrack}_{{\xi = \frac{x}{\lambda \; f}},{\eta = \frac{y}{\lambda \; f}}}} \\{= {{\frac{1}{4}\left( {{\overset{\_}{E}}_{X} - {j{\overset{\_}{E}}_{Y}}} \right)^{j\frac{2\pi}{\lambda \; f}{\alpha {({x - y})}}}} +}} \\{{{\left( {{{- j}{\overset{\_}{E}}_{X}} + {\overset{\_}{E}}_{Y}} \right)^{{- j}\frac{2\pi}{\lambda \; f}{\alpha {({x - y})}}}} +}} \\{{{\left( {{j{\overset{\_}{E}}_{X}} + {\overset{\_}{E}}_{Y}} \right)^{j\frac{2\pi}{\lambda \; f}{\alpha {({x - y})}}}} +}} \\{{{\left( {{\overset{\_}{E}}_{X} - {j{\overset{\_}{E}}_{Y}}} \right)^{{- j}\frac{2\pi}{\lambda \; f}{\alpha {({x - y})}}}},}}\end{matrix}$

wherein Ē_(X) and Ē_(Y) are implicitly dependent on x and y, f is thefocal length of the objective lens 2424, and λ is the wavelength of theincident illumination. The intensity is calculated by taking theabsolute value squared of E_(L). Simplifying the expression with theStokes parameter definitions, combining terms into cosines and sines,and substituting the shear α from:

${\alpha \cong {\frac{m\; \lambda}{\Lambda}t}},$

produces the final intensity pattern on the FPA 2426:

${I\left( {x,y} \right)} = {{\frac{1}{2}{S_{0}\left( {x,y} \right)}} + {\frac{1}{2}{S_{3}\left( {x,y} \right)}{\cos \left( {2\pi \frac{2{mt}}{f\; \Lambda}x} \right)}} + {\frac{1}{4}{{S_{2}\left( {x,y} \right)}\left\lbrack {{\cos \left( {2\pi \frac{2{mt}}{f\; \Lambda}\left( {x - y} \right)} \right)} - {\cos \left( {2\pi \frac{2{mt}}{f\; \Lambda}\left( {x + y} \right)} \right)}} \right\rbrack}} + {\frac{1}{4}{{{S_{1}\left( {x,y} \right)}\left\lbrack {{\sin \left( {2\pi \frac{2{mt}}{f\; \Lambda}\left( {x - y} \right)} \right)} + {\sin \left( {2\pi \frac{2{mt}}{f\; \Lambda}\left( {x + y} \right)} \right)}} \right\rbrack}.}}}$

This configuration enables the measurement of all four Stokes parametersby isolating the various white-light spatial carrier frequencies U₁ andU₂, defined as

${U_{1} = {2\frac{mt}{f\; \Lambda}}},{U_{2} = {2\sqrt{2}{\frac{mt}{f\; \Lambda}.}}}$

Additional Examples

The examples above are representative only and are selected for purposesof illustration. In other examples, the same or different combinationsof polarization parameters such as Stokes parameters can be estimated,and interferometers that include additional reflective surfaces and/orpolarization diffraction gratings can be used. Some examples aredescribed with respect to linear polarizers, but in other examples,circular polarizers can be used. In view of the many possibleembodiments to which the principles of the disclosed technology may beapplied, it should be recognized that the illustrated embodiments areonly preferred examples and should not be taken as limiting. I claim asmy invention all that comes within the scope and spirit of the appendedclaims.

1. An apparatus, comprising: at least one polarizing grating configuredto produce a dispersion compensated shear between portions of the aninput light flux associated with first and second polarizations; adetector situated to receive an output light flux corresponding to acombination of the sheared first and second portions of the input lightflux and produce an image signal; and an image processor configured toproduce a polarization image based on the image signal.
 2. The apparatusof claim 1, further comprising a polarization analyzer situated betweenthe at least one grating and the detector.
 3. The apparatus of claim 2,wherein the at least one polarizing grating includes two polarizinggratings, wherein the first grating is situated to produce a first shearportion by directing the first polarization along a first direction andthe second polarization along a second direction, and the second gratingis situated to produce a second shear portion by directing the firstpolarization directed by the first grating along the second directionand the second polarization directed by the first grating along thefirst direction.
 4. The apparatus of claim 3, wherein the firstpolarization and the second polarizations are orthogonal linearpolarizations or left and right circular polarizations.
 5. The apparatusof claim 3, wherein the at least one polarizing grating includes a firstpair and a second pair of polarizing gratings configured to producedispersion compensated shear along a first axis and a second axis. 6.The apparatus of claim 3, wherein the detector is an array detector. 7.The apparatus of claim 6, wherein the image processor is configured toproduce the polarization image based on an amplitude modulation ofinterference fringes.
 8. The apparatus of claim 7, wherein the imageprocessor is configured to select at least one spatial frequencycomponent of the recorded image signal and determine an imagepolarization characteristic based an intensity modulation associatedwith an image signal variation at the selected spatial frequency.
 9. Theapparatus of claim 8, wherein the image polarization characteristic isone or more or a combination of Stokes parameters S₀, S₁, S₂, and S₃.10. The apparatus of claim 3, wherein the polarizing gratings are blazedbirefringent gratings.
 11. The apparatus of claim 3, wherein thepolarizing gratings are liquid crystal gratings.
 12. The apparatus ofclaim 3, wherein the dispersion compensated shear is proportional to aseparation between the first grating and the second grating.
 13. Theapparatus of claim 3, wherein the first grating is situated to directthe first polarization above and away from the optical axis and thesecond polarization in a below and away from the optical axis, and thesecond grating is configured to direct the first polarization and thesecond polarization back towards the optical axis so as to produce thedispersion compensated shear.
 14. The apparatus of claim 3, wherein thefirst grating and the second grating have different or identical gratingperiods.
 15. The apparatus of claim 3, wherein the polarization image isa two dimensional image.
 16. The apparatus of claim 3, wherein shear fora spectral component of the input optical flux is proportional to awavelength associated with the spectral component.
 17. A method,comprising: receiving an input optical flux; producing a shear betweenfirst and second portions of the input optical flux associated withfirst and second states of polarization that is proportional to awavelength of the input optical flux by directing the first and secondportions to a pair of polarizing gratings; and estimating a polarizationcharacteristic of the input optical flux based on a spatial frequencyassociated with the shear in an intensity pattern obtained by combiningthe sheared first and second portions of the input optical flux.
 18. Themethod of claim 17, further comprising diffracting each of the first andsecond portions of the incident optical flux at the at least onediffraction grating so as to produce a shear having a magnitudeassociated with a grating period a wavelength associated with the inputoptical flux.
 19. The method of claim 18, wherein the shear is inverselyproportional to a grating period and directly proportional to a gratingorder.
 20. The method of claim 17, further comprising combining thefirst and second portions with at least one focusing optical element offocal length f, wherein the spatial frequency is inversely proportionalto f.
 21. An imaging polarimeter, comprising: a first polarizing gratingconfigured to diffract portions of an input light flux having a firststate of polarization and a second state of polarization in a firstdirection and a second direction, respectively; a second polarizinggrating configured to receive the diffracted portion from the firstpolarizing grating and diffract the portions associated with the firststate of polarization and the second state of polarization along thesecond direction and the first direction, respectively, so that thefirst and second portions propagate displaced from and parallel to eachother; a polarization analyzer configured to produce a common state ofpolarization of the first and second portions; and a focusing elementconfigured to combine the first and second portions; a detectorconfigured to receive the intensity pattern and produce a detectedintensity pattern.
 22. The polarimeter of claim 21, further comprisingan image processor configured to produce a polarization image based onthe detected intensity pattern.
 23. The polarimeter of claim 22, whereinthe detected intensity pattern is associated with a shear produced bythe displacement of the first and second portions.